Abstract
With this chapter, we come back to Élie Cartan's work by discussing his fundamental contributions in the realm of the integration theory of general systems of Pfaffian equations. Before turning to the analysis of Cartan's relevant works in the period 1899–1904, the historical context in which these achievements saw the light is taken into consideration. To this end, we discuss the pioneering works by Engel and von Weber that in a way laid the basis for the subsequent geometrical treatment provided by Cartan. The emergence of the notions of differential form and exterior derivative (bilinear covariant, in the language of that time) is analyzed, especially in view of Cartan's existence theorem.
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